"""JAM 5, line 5 instrument (founder, sharpening Grok's LINE 4): a
direct "distinct tuple count" for L>=8 is DEGENERATE, not new
information - the line-3 return-time probe already proved zero
collisions there, which by definition means every visited (L+1)-tuple
was distinct; counting them again just re-derives N-L+1. The
genuinely testable version of Grok's real question ("does the
recursion's coupling create detectable non-uniform structure?") is a
CONDITIONAL ENTROPY test on a small, fully-saturable state space:
for M small enough that M^2 << N (so every (a,b) pair gets many
samples), build the empirical distribution of Q(n) mod M given
(Q(n-2) mod M, Q(n-1) mod M). If Q were close to independent of its
own recent residues mod M, conditional entropy should sit near the
unconditional (marginal) entropy, log2(M) at max. Real structure
would show as conditional entropy well BELOW the marginal - some
(a,b) pairs sharply predicting the next residue."""

import math
from collections import defaultdict

N_MAX = 2_000_000
TEST_M = [54, 108, 162]


def build_q(n_max: int):
    Q = [0] * (n_max + 1)
    Q[1] = 1
    if n_max >= 2:
        Q[2] = 1
    for n in range(3, n_max + 1):
        Q[n] = Q[n - Q[n - 1]] + Q[n - Q[n - 2]]
    return Q


def entropy_bits(counts):
    total = sum(counts.values())
    h = 0.0
    for c in counts.values():
        p = c / total
        h -= p * math.log2(p)
    return h


def main():
    print("building Q(1..%d) ..." % N_MAX)
    Q = build_q(N_MAX)
    print("\n  M | marginal H (bits) | log2(M) | mean conditional H | cells | avg samples/cell | H-drop")
    for M in TEST_M:
        marg = defaultdict(int)
        joint = defaultdict(lambda: defaultdict(int))
        for n in range(3, N_MAX + 1):
            a, b, c = Q[n - 2] % M, Q[n - 1] % M, Q[n] % M
            marg[c] += 1
            joint[(a, b)][c] += 1
        H_marg = entropy_bits(marg)
        total_n = N_MAX - 2
        H_cond = 0.0
        for (a, b), dist in joint.items():
            n_ab = sum(dist.values())
            H_cond += (n_ab / total_n) * entropy_bits(dist)
        cells = len(joint)
        avg_samples = total_n / cells
        print("  %3d | %17.4f | %7.4f | %19.4f | %5d | %17.1f | %.4f"
              % (M, H_marg, math.log2(M), H_cond, cells, avg_samples, H_marg - H_cond))
    print("\nHonest reading: H-drop near 0 means conditioning on the last two")
    print("residues barely sharpens the next one - consistent with near-independence")
    print("at this resolution. A large H-drop would be real, cheap, detectable")
    print("evidence of structure, without needing any collision at all.")


if __name__ == "__main__":
    main()